The Methods of Maximum Flow and Minimum Cost
Flow Finding in Fuzzy Network
Alexandr Bozhenyuk1, Evgeniya Gerasimenko1, and Igor Rozenberg2
1
Southern Federal University, Taganrog, Russia
AVB002@yandex.ru, e.rogushina@gmail.com
2
Public Corporation “Research and Development Institute of Railway Engineers”, Moscow,
Russia
I.rozenberg@gismps.ru
Abstract. This article considers the problems of maximum flow and minimum
cost flow determining in fuzzy network. Parameters of fuzzy network are fuzzy
arc capacities and transmission costs of one flow unit represented as fuzzy tri-
angular numbers. Conventional rules of operating with fuzzy triangular num-
bers lead to a strong “blurring” of their borders, resulting in loss of self-
descriptiveness of calculations with them. The following technique of addition
and subtraction of fuzzy triangular numbers is proposed in the presented paper:
the centers are added (subtracted) by the conventional methods, and the borders
of the deviations are calculated using linear combinations of the borders of ad-
jacent values. The fact that the limits of uncertainty of fuzzy triangular numbers
should increase with the increasing of central values is taken into account. To
illustrate the proposed method numerical examples are presented.
Keywords: Fuzzy arc capacity, linear combination of borders, fuzzy triangular
number, fuzzy flow.
1 Introduction
This article deals with flow problems arising in networks. The network corresponds to
a directed graph G ( X , A), where X – the set of nodes, A – the set of arcs with
distinguished initial (source) and final (sink) nodes. Each arc ( xi , x j ) A has capaci-
ty determining the maximum number of flow units, which can pass along the arc. The
relevance of the tasks of maximum and minimum cost flow determining lies in the
fact that the researcher can effectively manage the traffic, taking into account the
loaded parts of roads, redirect the traffic, and choose the cheapest route. Suppose a
network, which arcs have capacities (qij ). Formulation of the problem of maximum
flow finding in the network is reduced to maximum flow determining that can be
passed along arcs of the network in view of their capacities [1]:
2 A. Bozhenyuk et al.
max,
x j Г( s )
sj kt
xk Г -1 (t )
, xi s,
ij ki , xi t ,
0, x s, t ,
(1)
x j Г ( xi ) xk Г 1 ( xi )
i
0 ij qij , ( xi , x j ) A.
In (1) ij – the amount of flow, passing along the arc ( xi , x j ) ; – the maximum
flow value in the network; s – initial node (source); t – final node (sink); qij – arc
capacity, Г ( x i ) – the set of nodes, arcs from the node xi X go to, Г 1 ( xi ) – the set of
nodes, arcs to the node xi X go from. ij represents, for example, the amount of
cars, going from the node xi X to the node x j X . The first equation of (1) defines
that we should maximize the total number of flow units emanating from the source
( ), which is equal to the total number of flow units entering the sink ( ). The se-
cond equation of (1) is a flow conservation constraint, which means that the total
number of flow units emanating from the source ( ) must be equal to the total num-
ber of flow units entering the sink ( ) and the total number of flow units emanating
from any node xi s, t must be equal to the total number of flow units entering the
node xi s,t. The third inequality of (1) is a bound constraint, which indicates that
the flow of value ij , passing along any arc ( xi , x j ) must not exceed its arc capacity.
The task of minimum cost flow determining in a network can be formulated as fol-
lows: suppose we have a network, which arcs have two associated numbers: the arc
capacity (qij ) and transmission cost (cij ) of one flow unit passing from the node
xi X to the node x j X . The essence of this problem is to find a flow of the given
value from the source to the sink, which doesn’t exceed the maximum flow in the
graph and has minimal transmission cost. In mathematical terms the problem of
minimum cost flow determining [2] in the network can be represented as follows:
c min,
( x i , x j ) A
ij ij
, xi s,
(2)
ij ki , xi t ,
x j Г ( xi ) x k Г 1 ( x i ) 0, x s, t ,
i
0 ij qij , ( xi , x j ) A.
Maximum Flow and Minimum Cost Flow Finding 3
In (2) cij – transmission cost of one flow unit along the arc ( xi , x j ), – given
flow value, that doesn’t exceed the maximum flow in the network.
In practice, the arc capacities, transmission costs, the values of the flow entering
the node and emanating from the node cannot be accurately measured according to
their nature. Weather conditions, emergencies on the roads, traffic congestions, and
repairs influence arc capacities. Variations in petrol prices, tolls can either influence
transmission costs. Therefore, these parameters should be presented in a fuzzy form,
such as fuzzy triangular numbers [3]. Thus, we obtain a problem statement of maxi-
mum and minimum cost flow problems in fuzzy conditions.
2 Literature Review of the Maximum and Minimum Cost Flow
Determining Tasks
The problem of the maximum flow finding in a general form was formulated by T.
Harris and F. Ross [4]. L. Ford and D. Fulkerson developed famous algorithm for
solving this problem, called “augmented path” algorithm [5]. Maximum flow problem
was considered in [1, 6].
There are different versions of the Ford-Fulkerson’s algorithm. Among them there
is the shortest path algorithm, proposed by J. Edmonds and R. Karp in 1972 [7], in
which one can choose the shortest supplementary path from the source to the sink at
each step in the residual network (assuming that each arc has unit length). The short-
est path is found according to the breadth-first search.
Determining the maximum flow in the transportation network in terms of uncer-
tainty has been studied less. In [8] a solution taking into account the interval capaci-
ties of arcs was proposed. S. Chanas [9] proposed to solve this problem by using so-
called “fuzzy graphs”. There are contemporary articles which solve the problem by
the simplex method of linear programming [10].
Many researchers have examined the task of minimum cost flow finding in crisp
conditions in the literature. Methods of its solution can be divided into graph tech-
niques and the methods of linear programming. In particular, solutions by the graph
methods are considered in [1, 6]. The advantages of this approach are great visualiza-
tion and less cumbersome. The minimum cost flow is proposed to find by Busacker-
Gowen and M. Klein’s algorithms in [2]. In [2, 6] a task of minimum flow determin-
ing is considered as a task of linear programming. This approach is cumbersome.
The methods of minimum cost flow finding in networks in fuzzy conditions can be
divided into two classes. The first class involves the use of conventional flow algo-
rithms for determining the minimum cost flow, which operate with fuzzy data instead
of crisp values and require cumbersome routines with fuzzy numbers. The second
class of problems implies the use of “fuzzy linear programming”, which was widely
reported in the literature [11, 12].
Authors [13] consider the tasks of “fully fuzzy linear programming”. These tasks
are cumbersome and can not lead to optimal solutions in the minimum cost flow de-
termination. The solution of fuzzy linear programming tasks by the comparison of
fuzzy numbers based on ranking functions is examined in [14].
4 A. Bozhenyuk et al.
3 Presented Method of Operating with Fuzzy Triangular
Numbers
Researcher is faced with the problem of fuzziness in the network, when considering
the problems of maximum and minimum cost flow finding. Arc capacities, flow val-
ues, passing along the arcs, transmission costs per unit of goods cannot be accurately
measured, so we will represent these parameters as fuzzy triangular numbers.
We will represent the triangular fuzzy numbers as follows: (a, , ), where a –
the center of the triangular number, – deviation to the left of the center, – devia-
tion to the right of center, as shown in Fig. 1.
( x)
1
α
0 a a1 ( ) a a2 ( ) a x
Fig. 1. Fuzzy triangular number.
Conventional operations of addition and subtraction of fuzzy triangular numbers
are as follows: let A1 and A2 be fuzzy triangular numbers, such as A1 a1, 1, 1
~ ~ ~
and A2 a2 , 2 , 2 . Therefore, the sum of triangular numbers can be written as:
~
A1 A2 a1 a2 , 1 2 , 1 2
~ ~
and the difference represented as:
A1 – A2 a1 a2 , 1 2 , 1 2 [3]. The disadvantage of the conventional meth-
~ ~
ods of addition and subtraction of fuzzy triangular numbers is a strong “blurring” of
the resulting number and, consequently, the loss of its self-descriptiveness. For exam-
ple, when adding the same triangular number with itself, the borders of its uncertainty
increase: (2,1,1) + (2,1,1) = (4, 2, 2) and (2,1,1) + (2,1,1) + (2,1,1) = (6, 3, 3). Gener-
ally, it is not true, because the center of the triangular number should increase, while
its borders must remain constant. The fact that the degree of borders “blurring” of
fuzzy number depends on the size of its center is not usually considered, when speci-
fying the triangular fuzzy numbers. Therefore, the more the center, the more “blurred”
the borders should be (while measuring 1 kg of material, we are talking “about 1 kg”,
implying the number “from 900 to 1100 g”, but while measuring 1 t. of material, im-
ply that “about 1 t.” is the number “from 990 kg to 1110 kg”).
Comparison of fuzzy triangular numbers according to various criteria is also very
difficult and time-consuming. Consequently, following method is proposed to use
Maximum Flow and Minimum Cost Flow Finding 5
when operating with triangular fuzzy numbers. Suppose there are the values of arc
capacities, flows or transmission costs in a form of fuzzy triangular numbers on the
number axis. Then when adding (subtracting) the two original triangular fuzzy num-
bers their centers will be added (subtracted), and to calculate the deviations it is nec-
essary to define required value by adjacent values. Let the fuzzy arc capacity (flow or
transmission cost) “near ~x ' ” is between two adjacent values “near ~x1 ” and “near ~x2 ”,
( x1 x ' x2 ) which membership functions ~x1 ( x1) and ~x2 ( x2 ) have a triangular
form, as shown in Fig. 2.
~x1
~x ' ~x 2
1
x
x1 x' x2
l1L l1R lL lR l 2L l 2R
Fig. 2. Given values of arc capacities (flows or transmission costs).
Thus, one can set the borders of membership function of fuzzy arc capacity (flow
or transmission cost) “near ~x ' ” by the linear combination of the left and right borders
of adjacent values:
( x2 x ) ( x x) L
lL l1L 1 2 l2 ,
( x2 x1 ) ( x2 x1 )
(3)
( x x) ( x x) R
lR 2 l1R 1 2 l2 .
( x2 x1 ) ( x2 x1 )
In (3) l L is the left deviation border of required fuzzy number, l R is the right de-
viation border. In the case when the central value of triangular number resulting by
adding (subtracting) repeats the already marked value on the number axis, its devia-
tion borders coincide with the deviation borders of the number marked on the number
axis. If required central value is not between two numbers, but precedes the first
marked value on the number axis, its deviation borders coincide with those of the first
marked on the axis. The same applies to the case when the required central value
follows the last marked value on the axis.
6 A. Bozhenyuk et al.
4 Solving the Task of Maximum Flow Finding in Fuzzy Network
The task of maximum flow finding in fuzzy network can be formulated as follows:
~ ~
~ sj kt max,
x jГ( s ) xk Г-1 (t )
~, xi s,
(4)
~ ~
ij ki ~, xi t ,
x jГ ( xi ) xk Г 1 ( xi ) ~
0, xi s, t ,
~ ~ ~
0 q , ( x , x ) A.
ij ij i j
~
In (4) ~ is required maximum fuzzy flow value in the network; ij – fuzzy
amount of flow, passing along the arc ( xi , x j ); q ~ – fuzzy capacity of the arc
ij
~
( xi , x j ); 0 is fuzzy number of the form (0, 0, 0), as it reflects the absence of the flow.
Let’s consider an example, illustrating the solution of this problem, represented in
Fig. 3. Let network, representing the part of the railway map, is given in a form of
fuzzy directed graph, obtained from GIS “Object Land” [15, 16]. Let the node x1 is a
source, node x12 is a sink. The values of arc capacities in the form of fuzzy triangular
numbers are defined above the arcs. It is necessary to calculate the maximum flow
value between stations “Kemerovo” ( x1 ) and “Novosibirsk-Gl.” ( x12 ) according to
Edmonds-Karp’s algorithm [7] and the method, described for operations with fuzzy
triangular numbers. Determining of maximum flow is based on sending flows along
the arcs of the network until one cannot send any additional unit of flow from the
source to the sink. Edmonds-Karp’s algorithm represents the choice of the shortest
supplementary path from the source to the sink at each step in the residual network
(assuming that each arc has unit length). Fuzzy residual network contains the arcs of
~
the form ( xi , x j ) with the fuzzy residual arc capacity q~ij ij , if the arcs ( xi , x j )
~
have the flow value q~ in the initial network; and the arcs of the form ( x , x )
ij ij j i
~ ~ ~
with the residual arc capacity ij , if the arcs ( xi , x j ) have the flow value ij 0 .
The arc ( xi , x j ) is called “saturated” when the flow, passing along it, equals to arc
capacity q~ . Other words, residual arc capacity defines how many flow units can be
ij
sent along the arc ( xi , x j ) to reach arc capacity. Residual arc capacity of arc saturat-
ed arc ( xi , x j ) is zero.
Maximum Flow and Minimum Cost Flow Finding 7
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km
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149
p.
8, 8) x8
B-
(45, (30, 5, 6)
r
ku
)
,2
So
2
) x9 (16, 2, 2)
(28, 5, 5
6,
t
II
na
(1
ga
bi
l.
x11
-G
Ur
m
(32,
ko
k
7 ) 6, 7
irs
6, ) x7
ed
,
sib
o
(32
Pr
( 2 ovo
ov
8)
(16
5)
vo
, 3)
er
, 3)
8,
n
5,
No
,2
em
ki
ha
5,
x12
8,
(20, 2
p
(20, 2
,2
x3
(4
Is
K
To
)
(25, 5, 4)
( 45
x5 x4 x1
,8
,8
3)
4)
(4
)
2,
,
0,
,4
0,
x10
7,
2
(2
(32, 6, (2
7)
7)
a
x2
ay
x6 rt.
sk
vo-So
q~ij
In
ero
Kem
tn aya
oek i j
Pr
Fig. 3. Initial network.
Therefore, the algorithm proceeds as follows: the first iteration of the algorithm
performs an augmenting chain x1x3 x8 x9 x11x12 . Push the flow, equals to (28, 5, 5)
units along it. The arc x9 , x11 becomes saturated, consequently, fuzzy residual ca-
pacity of the arc x9 , x11 equals to (0, 0, 0). Let’s define the fuzzy residual capacities
of the remaining arcs of augmenting chain. The arc x1, x3 has fuzzy residual capaci-
ty equals to (45, 8, 8) – (28, 5, 5). Thus, the central value of the resulting number is
17. It is located between adjacent arc capacities: (16, 2, 2) and (20, 2, 3) of the origi-
nal graph as shown in Fig. 4.
( x)
1
0 14 16 17 18 20 23 x
Fig. 4. Fuzzy triangular number with a center equals to 17 and its adjacent numbers.
Compute the left and the right deviation borders of the fuzzy triangular number
with a center of 17 according to (3). Thus, we obtain a fuzzy triangular number of the
form (17, 2, 2.25) units.
8 A. Bozhenyuk et al.
Fuzzy residual capacity of the arc ( x3 , x8 ) is (30, 5, 6) – (28, 5, 5). Consequently,
we obtain a fuzzy triangular number with a center of 2, located to the left of fuzzy
triangular number of the form (16, 2, 2). Deviation borders of the required number
coincide with deviation borders of the number (16, 2, 2). Thus, we obtain a fuzzy
triangular number of a type (2, 2, 2) units.
Fuzzy residual capacity of the arc ( x8 , x9 ) equals to (17, 2, 2.25) units, similarly
with the arc x1 , x3 .
Finally, fuzzy residual capacity of the arc x11, x12 is equal to (32, 6, 7) – (28, 5,
5), i.e. we obtain fuzzy number (4, 2, 2) units and fuzzy residual capacity of the arc
x12 , x11 equals to (28, 5, 5) units. Fuzzy residual capacities of the arcs
( x3 , x1 ), ( x8 , x3 ), ( x9 , x8 ), ( x11, x9 ), ( x12, x11) are (28, 5, 5) units.
The second iteration of the algorithm gives the augmenting chain
x1x2 x4 x5 x6 x10 x12 . Push the flow equals to (20, 2, 3) units along it. The arc x1 , x2
becomes saturated, consequently, fuzzy residual capacity of the arc x1 , x2 equals to
(0, 0, 0). Fuzzy residual capacity of the arc x2 , x4 is (40, 7, 7) – (20, 2, 3), i.e. we
obtain a fuzzy triangular number (20, 2, 3) units. Fuzzy residual capacity of the arc
x4 , x5 is the difference between the numbers (25, 5, 4) and (20, 2, 3). Thus, we get
a number with a center of 5, located to the left of the number (16, 2, 2), i.e. (5, 2, 2)
units. Fuzzy residual capacity of the arc x5 , x6 is equal to (22, 4, 4) – (20, 2, 3), i.e.,
(2, 2, 2) units. Fuzzy residual capacity of the arc x6 , x10 is equal to (32, 6, 7) – (20,
2, 3), i.e. (12, 2, 2) units. Fuzzy residual capacity of the arc x10 , x12 is (45, 8, 8) –
(20, 2, 3), i.e., (25, 5, 4) units. Fuzzy residual capacities of the arcs
( x2 , x1 ), ( x4 , x2 ), ( x5 , x4 ), ( x6 , x5 ), ( x10, x6 ), ( x10, x12 ) are (20, 2, 3) units.
The third iteration of the algorithm performs the augmenting chain
x1x3 x4 x5 x6 x10 x12 . Push the flow equals to (2, 2, 2) units along. The arc x5 , x6
becomes saturated. Let’s define fuzzy residual capacities of the remaining arcs of the
augmenting chain. Fuzzy residual capacity of the arc x1, x3 is (17, 2, 2.25) – (2, 2,
2), i.e. (15, 2, 2) units. Fuzzy residual capacity of the arc x3 , x1 is (30, 5, 6) units.
Fuzzy residual capacity of the arc x3 , x4 is equal to (28, 5, 5) – (2, 2, 2). We get the
number with a center of 26, located between adjacent values (25, 5, 4) and (28, 5, 5).
Compute the left and the right deviation borders of the fuzzy triangular number
with a center of 26 according to (3). Thus, we obtain a fuzzy triangular number of the
form (26, 5, 4.33) units.
Fuzzy residual capacity of the arc x4 , x5 is equal to (5, 2, 2) – (2, 2, 2), i.e. (3, 2,
2) units. Fuzzy residual capacity of the arc x6 , x10 is equal to (12, 2, 2) – (2, 2, 2),
i.e. (10, 2, 2) units. Fuzzy residual capacity of the arc x10 , x12 is equal to (25, 5, 4) –
(2, 2, 2), i.e. we obtain a fuzzy number with a center of 23, located between adjacent
values (22, 4, 4) and (25, 5, 4), therefore, the left deviation border of the number with
a center of 23 equals to 4.33, the right deviation border is 4. We obtain fizzy triangu-
Maximum Flow and Minimum Cost Flow Finding 9
lar number (23, 4.33, 4) units. Fuzzy residual capacities of the arcs
( x5 , x4 ), ( x6 , x5 ), ( x10, x6 ), ( x10, x12 ) are (22, 4, 4) units.
After execution of three iterations of the algorithm it is impossible to pass any sin-
gle additional flow unit. The total flow is (28, 5, 5) + (20, 2, 3) + (2, 2, 2) units.
Therefore, we obtain a fuzzy triangular number with a center of 50, located to the
right of the number (45, 8, 8) with the borders, repeated deviations of the number 45:
(50, 8, 8) units.
Thus, the maximum flow value between the stations “Kemerovo” and “Novosi-
birsk-Gl.” is (50, 8, 8) units. Let us carry out an interpretation of the results: the max-
imum flow between the given stations can not be less than 42 and more than 58 units,
with the highest degree of confidence it will be equal to 50 units. But with changes in
the environment, repairs on the roads, traffic congestions the flow is guaranteed to lie
in the range from 42 to 58 units. Fuzzy optimal flow distribution along the arcs and
labels of the nodes is shown in Fig. 5. Saturated arcs are bold.
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km
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49
(+x7, (5, 2, 2))
1
p.
5) x8
B-
r
5, (28, 5, 5)
))
ku
(28,
l.
, 2.25
So
-G
t
x9
ina
(+x10, (2, 2, 2)) (28, 5, 5)
k
irs
(17, 2
mb
sib
II
x11 (+x7, (5, 2, 2))
ga
vo
vo
ko
2)
)
5) x7
Ur
No
ed
ero
,
(+x1 ,
2, ,5
Pr
,
)
(5 (28
,6
Kem
0,
(+x5, (5, 2, 2))
vo
,5
x1 )
no
(+ ,2
ki
( 30
,2
ha
p
x12 (2 x3 (+x1, ∞)
To
(22
Is
,4 (22, 4, 4)
,4 x5 x4 x1
)
)
,3
(2
x10
)
2
4
0,
0,
(+x4, (5, 2, 2))
4,
(2
2,
(22, 4
)) 2,
2,
a
, 4)
3)
25 ,
(2
2. , (17
ay
(+x6, (2, 2, 2))
sk
x2
x3
In
(+
.
x6 ort
ya
o-S (+x1, (20, 2, 3))
~
na
rov
ij
(+x5, (2, 2, 2)) me
kt
Ke
oe
Pr
i j
Fig. 5. Network with maximum flow of (50, 8, 8) units.
5 Solving the Task of Minimum Cost Flow Determining in Fuzzy
Network
Consider the problem of minimum cost flow finding in a network according to fuzzy
values of arc capacities, flows and transmission costs of one flow unit.
10 A. Bozhenyuk et al.
c~ min,
~
ij ij
( xi , x j ) A
~, xi s,
(5)
~ ~
ij ki ~, xi t ,
x j Г ( xi ) xk Г 1 ( xi ) ~
0 , xi s , t ,
~ ~ ~
0 q , ( x , x ) A.
ij ij i j
In (5) c~ij – fuzzy transmission cost of one flow unit along the arc ( xi , x j ), ~ –
given fuzzy flow value, that doesn’t exceed the maximum flow ~ in the network.
Let us turn to the graph, shown in Fig. 3. Fuzzy values of transmission costs in ad-
dition to fuzzy arc capacities are given in this task:
с~x1x2 (12, 3, 3); с~x1x3 (6,1, 2); с~x3 x4 (10, 2, 3); с~x2 x4 (18, 4, 5); с~x3 x8 (4,1,1);
с~x4 x5 (12, 3, 3); с~ (20, 5, 6); с~
x5 x7 (15, 4, 4); с~
x8 x7 (15, 4, 4); с~ x7 x8 (21, 6, 7);x8 x9
с~x7 x9 (10, 2, 3); с~x5 x6 (30, 8, 9); с~x6 x10 (8, 2, 2); с~x9 x11 (19, 5, 5);
с~
x11x10 (32, 7,12); с~ (32, 7,12); с~
x10 x11 (25, 7, 8); с~
x11x12 (20, 5, 6).x10 x12
It is necessary to find a flow value ~ of (45, 8, 8) units from the source to the sink,
which has a minimal cost. Consider the Busacker-Gowen’s [2] algorithm, taking into
account the fuzzy capacities and costs to solve this problem:
Step 1. Assign all arc flows and the flow rate equal to zero.
Step 2. Determine the modified arc costs с~ * that depend on the value of the al-
ij
ready found flow as follows:
c~ij , if ~ ~
0 ij q~ij ,
~
c~ij* , if ij q~ij ,
~ ~
c ji , if ij 0.
Step 3. Find the shortest chain (in our case – the chain of minimal cost) [2] from
the source to the sink taking into account arc costs с~ * , found in the step 1. Push the
ij
flow along this chain until it ceases to be the shortest. Receive the new flow value by
adding the new flow value, passing along the considered chain, to the previous one. If
the new flow value equals to ~ , then the end. Otherwise, go to the step 2.
Solve this problem, taking into account fuzzy arc capacities costs.
~
Step 1. Assign all ij 0.
Step 2. Determine c~ij* c~ij .
Step 3. Find the shortest path by the Ford’s algorithm [1]: x1x3 x8 x9 x11x12 of the
total cost of (75, 8, 8) standard units. Push the flow, equals to (28, 5, 5) units along
this chain.
Maximum Flow and Minimum Cost Flow Finding 11
Step 2. Define the new modified fuzzy arc costs:
с~x*1 x3 (6,1, 2); с~x*3 x1 – (6,1, 2); с~x*3 x8 (4,1,1); с~x*8 x3 – (4,1,1); с~x*8 x9 (21, 6, 7);
с~x*9 x8 – (21, 6, 7); с~x*9 x11 ; с~x*11 x9 – (19, 5, 5); с~x*11 x12 (25, 7, 8); с~x*12 x11 – (25, 7, 8).
Step 3. Find
the shortest path using the obtained modified costs:
x1 x3 x4 x5 x6 x10 x12 of the total cost of (86, 8, 8) standard units. Push the flow, equals
to (17, 2, 2.25) units along this chain. As a result, we obtain the total flow equals to
(45, 8, 8) units, having a total transmission cost along the network, equals to (28, 5, 5)
× ((75, 8, 8) + (86, 8, 8)) = (3562, 8, 8) standard units. There are fuzzy flow values
~ ~
ij under the arcs and fuzzy transmission costs c~ij of optimal fuzzy flow values ij
above the arcs of the graph, saturated arcs are bold as shown in Fig. 6.
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(28, 5,5)
p.
5, 5) x8
B-
(28, (112, 8, 8)
t
r
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8, 8)
(588,
bi
So
5) x9
om
(28, 5,
l.
-G
k
II
k
ed
8 , 8)
irs
x11 (532 ,
ga
Pr
5) x7
sib
, 5,
Ur
(28 8, 8)
vo
x3 (4
o
,
No
5)
0
ov
( 1 ovo
(70 5,
2. 8
er
x12 ( (2
70 , 8 )
n
2,
ki
em
)
ha
,8
17
p
7,
,8
Is
K
To
,8
,2 ,8
,2
70
(3 (17, 2, 2.5) )
4 .5
(1
0, ) x5 x4 x1
8,
8) (204, 8, 8)
x10
, 8 5)
10 , 2.
(17,
)
,8
,2
a
2, 2.5
ay
7
( 136 )
(1
sk
, 8, 8 x2
In
rt.
(5
)
~
ij
So
x6
o-
ov
ya
er
na
i ~ j
em
c ( ij )
kt
oe
K
Pr
Fig. 6. Network with the flow of (45, 8, 8) units and transmission costs of each arc of the total
cost (3562, 8, 8) standard units.
6 Conclusion
This paper examines the problems of maximum and minimum cost flow determining
in networks in terms of uncertainty, in particular, the arc capacities, as well as the
transmission costs of one flow unit are represented as fuzzy triangular numbers. The
technique of addition and subtraction of triangular numbers is considered. Presented
technique suggests calculating the deviation borders of fuzzy triangular numbers
based on the linear combinations of the deviation borders of the adjacent values. The
fact that the limits of uncertainty of fuzzy triangular numbers should increase with the
increasing of central values is taken into account. Advantage of the proposed method
lies in the fact that operations with fuzzy triangular numbers don’t lead to a strong
12 A. Bozhenyuk et al.
“blurring” of their deviation borders, it makes calculations with such numbers more
effective.
Acknowledgments. This work has been supported by the Russian Research project
№ 11-01-00011а.
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